Some upper bounds for 3-rainbow index of graphs

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G , denoted by rxk(G). In this paper, we consider 3-rainbow index rx3(G) of G. We first show that for connected graph G with minimum degree δ(G) ≥ 3, the tight upper bound of rx3(G) is rx3(G[D]) + 4, where D is the connected 2-dominating set of G. And then we determine a tight upper bound for Ks,t(3 ≤ s ≤ t) and a better bound for (P5, C5)-free graphs. Finally, we obtain a sharp bound for 3-rainbow index of general graphs.